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eduKateSG · Why Mathematics? · Log-Rank Tests for Reliability Curves: Risk Sets, Expected Failures, Censoring and Proportional-Hazard Limits
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes practical when two reliability curves must be compared across their full observed follow-up rather than at one chosen milestone. The log-rank test repeatedly compares observed failures with the number expected from each risk set under equal hazards, then combines those deviations with their covariance into a test statistic.
This guide develops risk sets, observed-minus-expected counts, variance, censoring, stratification, the link to Cox regression and the proportional-hazards setting in which the ordinary log-rank test is most sensitive. A p-value is not an effect size, crossing curves can cancel, and a non-significant result does not prove engineering equivalence.
Full chapter index
Chapters 1–5 · Understand the question
Chapters 6–10 · Reproduce the example
Chapters 11–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–25 · Study and verify
CHAPTER 1 OF 25 · Understand the question
1. Why compare complete reliability curves
A milestone comparison can miss differences before or after the selected time. Two designs may both have 90% survival at 1000 hours but separate strongly at 400 hours. This is the central mathematical mechanism in why compare complete reliability curves. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A global test can still hide when the difference occurs. Plot curves and risk sets first. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind state the null hypothesis. The usual two-sample log-rank null says the event hazards are equal over time between groups under the sampling framework. In the illustrative case, If the groups truly share hazards, labels should not systematically predict failures within risk sets. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Failure to reject is not proof of identical curves. Write the null and alternative.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat risk sets at each event time as a model-checking problem rather than a vocabulary exercise. Just before each distinct failure time, count how many observed units remain at risk in each group. At 500 hours, group A may have 18 at risk and group B 12. Recompute the example from the raw values and label every unit. Censored units leave later risk sets but are not counted as failures. Build an event-time table.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of observed failures becomes useful only when its mechanism is explicit. At each event time, record the total failures and how many occurred in each group. Consider the worked illustration: If two of three tied failures are in A, A observed count is two. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Timestamp rounding can alter tie structure. Preserve event resolution.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://pubmed.ncbi.nlm.nih.gov/5910392/. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use expected failures under the null to practise sensitivity analysis. For one event with n_A of n total at risk, A’s expected count is n_A/n; with d tied events it is d n_A/n. With 18 of 30 at risk and two failures, A expects 1.2 under the null. Before accepting the result, identify the quantity that would change the conclusion most. Expected count is a conditional calculation, not a forecast for one unit. Keep the denominator at the event time.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for observed minus expected score. Sum O_Aj-E_Aj across event times; positive values mean more observed A failures than expected under the chosen coding. The initial example is: Contributions 0.4, -0.2 and 0.3 sum to 0.5. Then create a nearby case where the usual shortcut becomes unreliable. The sign reverses when group labels reverse. Declare the reference group.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect hypergeometric variance to a real decision while protecting the boundary between calculation and judgement. With tied failures, the conditional allocation variance uses the finite risk-set hypergeometric form. The variance shrinks when a risk set is very unbalanced. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Using a simple binomial formula can be wrong for several tied events. Use the exact event-time expression.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
If summed O-E is 2.0 and its variance is 1.6, the squared standardised statistic is 2.5. For two groups it is compared with a one-degree-of-freedom chi-square approximation. This is the central mathematical mechanism in a complete worked statistic. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Small samples or few events may need exact or permutation approaches. Report counts and approximation. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind more than two groups. A vector of group scores and covariance matrix creates a chi-square test with usually g-1 degrees of freedom. In the illustrative case, Three designs can be compared in one global test. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A significant global test does not identify every differing pair. Plan multiplicity-aware follow-ups.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat right censoring as a model-checking problem rather than a vocabulary exercise. Independent or non-informative censoring within groups allows censored units to contribute risk time until censoring. A unit working at cutoff remains in risk sets up to that time. Recompute the example from the raw values and label every unit. Different condition-driven dropout can bias the comparison. Compare censoring patterns.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of tied event times becomes useful only when its mechanism is explicit. Standard software applies an event-time formula or approximation when failures share a timestamp. Consider the worked illustration: Daily recording can create many ties. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Different tie conventions can matter in coarse data. Document timestamp precision and method.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use stratified log-rank tests to practise sensitivity analysis. Stratification computes scores within nuisance strata and combines them, comparing groups within each stratum. Compare designs within site or load band. Before accepting the result, identify the quantity that would change the conclusion most. Stratification does not estimate a causal effect or fix unmeasured confounding. Predefine a small number of meaningful strata.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://doi.org/10.2307/2344317. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for connection with cox regression. The ordinary two-group log-rank test is closely related to the score test for a group indicator in a Cox proportional-hazards model. The initial example is: Both use risk-set comparisons under proportional hazards. Then create a nearby case where the usual shortcut becomes unreliable. The test alone provides no hazard-ratio estimate. Pair it with an effect estimate and interval.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
CHAPTER 14 OF 25 · Check assumptions
14. Why proportional hazards matters for power
Connect why proportional hazards matters for power to a real decision while protecting the boundary between calculation and judgement. The standard log-rank weighting is especially efficient for sustained proportional-hazard alternatives. A roughly constant hazard ratio gives contributions with consistent direction. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The test remains computable when hazards cross but may have weak power. Inspect proportionality.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Early harm and later benefit can create positive and negative O-E contributions that cancel. A redesign may fail more during burn-in but less during wear-out. This is the central mathematical mechanism in crossing reliability curves. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A large p-value can coexist with important time-specific differences. Use plots, RMST or planned alternatives. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind weighted log-rank tests. Weights can emphasise early, middle or late event times for a pre-specified alternative. In the illustrative case, A late-weighted test may target wear-out separation. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Choosing weights after seeing curves inflates false-positive risk. Pre-specify the scientific alternative.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat effect size versus significance as a model-checking problem rather than a vocabulary exercise. The log-rank statistic answers an evidence question; it does not quantify the magnitude in hours or probability points. A tiny persistent difference can be significant in a large fleet. Recompute the example from the raw values and label every unit. A small study can miss a meaningful effect. Report survival differences, RMST or hazard ratios.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of sample size depends on events becomes useful only when its mechanism is explicit. For common proportional-hazard planning, information is driven chiefly by the number of observed events, then allocation and effect assumptions. Consider the worked illustration: Long follow-up may be needed if failures are rare. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Optimistic event-rate assumptions underpower the study. Plan with sensitivity ranges.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use non-inferiority and equivalence to practise sensitivity analysis. Ordinary failure to reject a difference is not an equivalence conclusion; margin-based designs and confidence intervals are required. A new design may need to exclude more than a declared reliability loss. Before accepting the result, identify the quantity that would change the conclusion most. A margin chosen after data is not credible. Set the margin before observation.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for diagnostics and reporting. Show Kaplan–Meier curves, numbers at risk, censoring marks, event counts, the test method and effect estimates. The initial example is: A risk table reveals when late curves rest on few units. Then create a nearby case where the usual shortcut becomes unreliable. A p-value without follow-up support can overstate precision. Report the full evidence package.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect decision interpretation to a real decision while protecting the boundary between calculation and judgement. Combine statistical evidence with failure consequence, absolute reliability and engineering plausibility. A significant but tiny difference may not justify redesign. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The test does not optimise cost or certify safety. Translate effects into decision units.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Create two small censored groups, compute risk sets, expected counts, O-E contributions and the total statistic. Then swap labels and check that the squared statistic stays the same. This is the central mathematical mechanism in practice activity. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Toy data are for learning only. Audit every event time. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind parent and educator guidance. The log-rank test connects fractions, expected values, variance, conditional probability and graphs. In the illustrative case, Ask why expected failures change as risk sets shrink. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Begin with one event time and coloured tokens. Delay the chi-square step until the table is clear.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat common misconceptions as a model-checking problem rather than a vocabulary exercise. The log-rank test is not a test of medians alone, p>0.05 is not proof of equality and significance is not practical importance. Crossing curves can cancel contributions. Recompute the example from the raw values and label every unit. Censoring is not a failure event. Translate the statistic and its limits.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
Continue with the Mathematics Learning Hub, the eduKate Mathematics Learning System, the approach to learning mathematics, the Singapore Mathematics curriculum overview, the cross-subject learning approach and the previous reliability article. Each destination has its own role: subject navigation, learning method, school-to-advanced progression, or continuity with the latest reliability mechanism.
The mathematics of research and next reading becomes useful only when its mechanism is explicit. Mantel developed rank statistics for censored survival data, while Peto and Peto derived efficient rank-invariant procedures for censored group comparisons. Consider the worked illustration: Recreate the risk-set allocation logic before using a software p-value. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The method paper does not select an engineering margin. Read the original tests beside modern effect-size reporting.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Quick answers
What should I calculate first? Start with the estimand or failure event, then a simple baseline.
Does the advanced method guarantee accuracy? No. It needs input, numerical and model checks.
Which school mathematics matters? Algebra, graphs, probability, functions and careful units all contribute.
How can a student practise safely? Use fictional or open benchmark data and preserve every intermediate value.
What makes a career claim responsible? Describe how the mathematics is used without promising admission, salary or employment.
Extended practice laboratory
Practice case 1: Why compare complete reliability curves
Setup: Use this chapter’s illustrative case: Two designs may both have 90% survival at 1000 hours but separate strongly at 400 hours.
Task: Plot curves and risk sets first.
Checking target: A global test can still hide when the difference occurs.
Practice case 2: State the null hypothesis
Setup: Use this chapter’s illustrative case: If the groups truly share hazards, labels should not systematically predict failures within risk sets.
Task: Write the null and alternative.
Checking target: Failure to reject is not proof of identical curves.
Practice case 3: Risk sets at each event time
Setup: Use this chapter’s illustrative case: At 500 hours, group A may have 18 at risk and group B 12.
Task: Build an event-time table.
Checking target: Censored units leave later risk sets but are not counted as failures.
Practice case 4: Observed failures
Setup: Use this chapter’s illustrative case: If two of three tied failures are in A, A observed count is two.
Task: Preserve event resolution.
Checking target: Timestamp rounding can alter tie structure.
Practice case 5: Expected failures under the null
Setup: Use this chapter’s illustrative case: With 18 of 30 at risk and two failures, A expects 1.2 under the null.
Task: Keep the denominator at the event time.
Checking target: Expected count is a conditional calculation, not a forecast for one unit.
Practice case 6: Observed minus expected score
Setup: Use this chapter’s illustrative case: Contributions 0.4, -0.2 and 0.3 sum to 0.5.
Task: Declare the reference group.
Checking target: The sign reverses when group labels reverse.
Practice case 7: Hypergeometric variance
Setup: Use this chapter’s illustrative case: The variance shrinks when a risk set is very unbalanced.
Task: Use the exact event-time expression.
Checking target: Using a simple binomial formula can be wrong for several tied events.
Practice case 8: A complete worked statistic
Setup: Use this chapter’s illustrative case: For two groups it is compared with a one-degree-of-freedom chi-square approximation.
Task: Report counts and approximation.
Checking target: Small samples or few events may need exact or permutation approaches.
Practice case 9: More than two groups
Setup: Use this chapter’s illustrative case: Three designs can be compared in one global test.
Task: Plan multiplicity-aware follow-ups.
Checking target: A significant global test does not identify every differing pair.
Practice case 10: Right censoring
Setup: Use this chapter’s illustrative case: A unit working at cutoff remains in risk sets up to that time.
Task: Compare censoring patterns.
Checking target: Different condition-driven dropout can bias the comparison.